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# properties of ranks of matrices

Let $D$ be a division ring and $M$ a matrix over $D$. In this entry, the ranks of $M$ are defined, and the following properties regarding ranks and elementary matrix operations are shown in the table below:

preservation of ranks of $M$

operation | left row rank | right row rank | left column rank | right column rank |
---|---|---|---|---|

row exchange | yes | yes | yes | yes |

column exchange | yes | yes | yes | yes |

row addition | yes | yes | yes | yes |

column addition | yes | yes | yes | yes |

left non-zero row scalar multiplication | yes | no | no | yes |

left non-zero column scalar multiplication | no | yes | yes | no |

right non-zero row scalar multiplication | no | yes | yes | no |

right non-zero column scalar multiplication | yes | no | no | yes |

From the properties above, one sees that

left column rank Align=Align right row rank, $\qquad$ right column rank Align=Align left row rank.

If $D$ is a field, all ranks of $M$ are identical, since left and right multiplications are the same. Now, back to assuming $D$ a division ring. Refer the right ranks of $M$ to be either the right row rank or right column rank of $M$. Left ranks, row ranks, and column ranks are similarly referred. Ranks of $M$ can be either one of the four ranks of $M$.

The following are additional properties of ranks of matrices of $D$:

1. if $r\neq 0$, then $rM$ and $Mr$ have the same ranks as those of $M$.

More to come…

## Mathematics Subject Classification

15A03*no label found*

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